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We are asked to simplify the following expression:

\[
3x^2 \left( 2x^{-2}y^{-3}z^{-1} \right)^{-3}
\]

### Step-by-Step Solution:

1. **Simplify the inner expression:** 
   The inner expression is \( 2x^{-2}y^{-3}z^{-1} \), and it's raised to the power of -3.

2. **Apply the exponent to each factor in the parentheses:**
   Using the rule \( (ab)^n = a^n b^n \), we can distribute the exponent \(-3\) to each factor in the parentheses:
   
   \[
   (2x^{-2}y^{-3}z^{-1})^{-3} = 2^{-3} \cdot (x^{-2})^{-3} \cdot (y^{-3})^{-3} \cdot (z^{-1})^{-3}
   \]

3. **Simplify each term:**
   - \( 2^{-3} = \frac{1}{8} \)
   - \( (x^{-2})^{-3} = x^{6} \) (because \((-2) \cdot (-3) = 6\))
   - \( (y^{-3})^{-3} = y^9 \)
   - \( (z^{-1})^{-3} = z^3 \)

   So, the simplified expression becomes:
   
   \[
   \frac{1}{8} x^6 y^9 z^3
   \]

4. **Combine with the outer term \( 3x^2 \):**
   Now, multiply this result by \( 3x^2 \):
   
   \[
   3x^2 \cdot \frac{1}{8} x^6 y^9 z^3 = \frac{3}{8} x^{8} y^9 z^3
   \]

Thus, the simplified expression is:

\[
\boxed{\frac{3}{8} x^8 y^9 z^3}
\]

Would you like further details or have any additional questions?

Here are some related questions for practice:

1. How would you simplify \((5x^{-1}y^2)^{-2}\)?
2. What is the result of \(2x^3 (3x^{-4}y^5)^{2}\)?
3. How do you handle negative exponents when they appear in a fraction?
4. If \(a^n \cdot a^m = a^{n+m}\), how would you simplify \((x^3 \cdot x^{-5})\)?
5. What is the simplified form of \((3xy^{-1}z^2)^{-4}\)?

**Tip:** Always apply the power to each term inside the parentheses when dealing with expressions raised to a power.

Which expression is equivalent to 3x^2(2x^{-2}y^{-3}z^{-1})^{-3}?

Learn how to simplify the expression 3x^2(2x^{-2}y^{-3}z^{-1})^{-3}. This problem covers exponent rules, including handling negative exponents and distributing powers. Suitable for high school students in Grades 9-12.

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